2.2. Martingales

소챕터 2.1에서는 Filtration과 Counting Process에 대해 배웠다. 이번 소챕터에서는 Martingale에 대해 다룰 것이다. Martingale은 소위 말하면 random walk와 같은 모습을 보인다 (generalization of mean 0 random walk). 이를 이해하기 위해 Probability Theory에서의 conditional expectation에 대한 예습이 필요하다.

특히 4번이 매우 중요하다.

Definition (Martingale and Submartingale in descrete time)

The sequence of random variables and \(\sigma\)-fields \((X_n,\mathcal{F}_n)\), \(n=1,2,\ldots\) is a martingale if for each \(n\)

1. \(X_n\in \mathcal{F}_n\),
2. \(\mathcal{F}_n\subset\mathcal{F}_{n+1}\),
3. \(E(|X_n|)<\infty\),
4. \(X_n=E(X_{n+1}|\mathcal{F}_n)\) almost surely.

The sequence \((X_n,\mathcal{F}_n)\), \(n=1,2,\ldots\) is a submartingale if in (4) the equality is replaced by “\(\le\)”.

For a discrete-time martingale, we have \[E(X_n|\mathcal{F}_m)=X_m\mbox{ }\mbox{ }\mbox{ }\mbox{ for all }\mbox{ }m<n. \mbox{ }\mbox{ }\mbox{ }\mbox{ }\mbox{ (이를 condition 4.2 라고 하자) }\]

• 1번과 adapted(to the Filtration)의 정의로 \(\mathcal{F}_n\)은 only upto time \(n\) 까지 정보를 갖는다는 의미이다 (이전 2.1에서 \(U(t)\in\mathcal{F}_t\)를 통해 설명했다).

• 2번은 increasing의 의미이다.

• 4번이 매우 중요하다. 그 의미는 말하자면 “Information for tomorrow with information for now is the information for now, e.g., just average by now”이다.

• 또한, You are doing better in submartingales than martingales (Submatingale은 시간에 대해 nondecreasing한 성격을 가진다).

• \(4\iff 4.2\)를 증명하기 위해 conditional expectation에 대한 Theorem을 이해해야 한다 (probability theory 9-2 참고): if \(\mathcal{G}_1,\mathcal{G}_2\subset\mathcal{F}\), \(\mathcal{G}_1 \subset \mathcal{G}_2\)이고 \(X\)가 random variable일 때 \[ E(E(X|\mathcal{G}_1)|\mathcal{G}_2) =E(X|\mathcal{G}_1) = E(E(X|\mathcal{G}_2)|\mathcal{G}_1). \]

  • e.g) Let \(\mathcal{G}\subset\mathcal{F}\). Let \(Z\) be the random variable such that \(Z:\Omega\rightarrow \mathbb{R}\). Let \(L^2=\{\mbox{vector space spanned by all r.v } Z \mbox{ such that } E(Z^2)<\infty\}\) (Hilbert space). Let \(V=\{\mbox{space of all r.v's which are } \mathcal{G} \mbox{-measurable in } L^2 \}\). Then, \[ E(X|\mathcal{G})=\prod_V X. \]

  • 아래 그림 참고(notation은 좀 다르다).

    

• 4 \(\implies\) 4.2 : \(E(X_n|\mathcal{F}_{n-2})=E(E(X_n|\mathcal{F}_{n-1})|\mathcal{F}_{n-2})=E(X_{n-1}|\mathcal{F}_{n-2})=X_{n-2}.\) Let \(m=n-2\) (done).

예제 1 (Random walk)

Let \(\xi_1,\xi_2,\ldots\) i.i.d with mean 0. Let \(X_n=\sum_{i=1}^n \xi_i\), \(\mathcal{F}_n=\sigma(\xi_1,\ldots,\xi_n)\). The series \(X_n\) is called a random walk. Note that \(\mathcal{F}_n=\sigma(\xi_1,\ldots,\xi_n)\) means in \((\Omega,\mathcal{F},P)\), \(X_1,\ldots,X_n\in\mathcal{F}_n\), \(X_{n+1}\notin \mathcal{F}_n\).

• Claim: \((X_n,\mathcal{F}_n)\) is a martingale

• Proof: Condition 1,2,3 is trivial. For proving 4, note that \[ E(X_{n+1}|\mathcal{F}_n)=\sigma(\xi_1,\ldots,\xi_n))=E(\xi_{n+1}+X_n|\xi_1,\ldots,\xi_n)=X_n+E(\xi_{n+1})=X_n. \]

예제 2

Let \(\{\mathcal{F}_n,n=1,2,\ldots\}\) be a filtration, and let \(Y\) be an integrable random variable. Then, \(E(Y|\mathcal{F}_n)\) is a martingale.

• Claim: Let \(X_n=E(Y|\mathcal{F}_n)\). Then, \((X_n,\mathcal{F}_n)\) is a martingale.

• Proof: Condition 1 is true by the definition of conditional expectation (\(X_n=E(Y|\mathcal{F}_n)\) is \(\mathcal{F}_n\)-measurable, i.e., \(X_n\in\mathcal{F}_n\)). Condition 2 is true because \(\{\mathcal{F}_n,n=1,2,\ldots\}\) is filtrations. Condition 3 is such that \(E(|X_n|)=E(E(|Y||\mathcal{F}_n))= E(|Y|)<\infty\). Condition 4 can be proved \[ E(X_{n+1}|\mathcal{F}_n)=E(E(Y|\mathcal{F}_{n})|\mathcal{F}_{n+1})= E(Y|\mathcal{F}_{n})=X_n. \]

Definition (Martingale and Submartingale in continuous time)

Let \(\mathcal{T}\) be an interval of the form \([0,\tau)\) or \([0,\tau]\), where \(\tau\) is finite. Given a filtration \(\{\mathcal{F}_t,t\in\mathcal{T}\}\), a martingale is a process \(M(\cdot)\) which is “cadlag” such that;

• \(M(\cdot)\) is adapted,
• \(M(\cdot)\) is integrable, i.e., \(E(|M(t)|)<\infty\) for all \(t\in \mathcal{T}\),
• \(M(\cdot)\) satisfies the martingale property such that \[ E(M(t)|\mathcal{F}_s)=M(S)\mbox{ }\mbox{ }\mbox{ }\mbox{ for all }\mbox{ }s<t. \] The process is called the submartingale if, as before, the equality is replaced by an inequality. The martingale is called square integrable if \[\sup_{t\in\mathcal{T}}E(M(t)^2)<\infty\].

• a function is “cadlag” if it is continuous from the right, and has the left limit (cadlag 조건을 거는 이유는 \(M(\cdot)\)이 어떠한 small interval에서 too wiggly함을 방지하기 위함이다).

• e.g) Is \(N_i(t)=I(X_i\le t)\) martingale? No, by the martingale property.

Lemma 1

If \(\{(X_n,\mathcal{F}_n), n=1,2,\ldots\}\) is a submartingale, and if \(\varphi\) is a nondecreasing convex function such that \(\varphi(X_n)\) is integrable for all \(n\), then \(\{(\varphi(X_n),\mathcal{F}_n),n=1,2,\ldots\}\) is again a submartingale.

Also, if \(\{(X_n,\mathcal{F}_n), n=1,2,\ldots\}\) is a martingale, then we do not need \(\varphi\) to be nondecreasing.

• Proof (A): start with \(X_n\le E(X_{n+1}|\mathcal{F}_n)\) (미래가 현재보다 낫다 \(\leftarrow\) submartingale의 정의). Apply \(\varphi\) to both sides such that \[\begin{eqnarray*} \varphi(X_n)&\le& \varphi(E(X_{n+1}|\mathcal{F}_n))\mbox{ }\mbox{ }\mbox{ }(\varphi\mbox{ is nondecreasing})\\ &\le& E(\varphi(X_{n+1}|\mathcal{F}_n)).\mbox{ }\mbox{ }\mbox{ }(\mbox{Jensen's Inequality}) \end{eqnarray*}\]

• Proof (B): start with \(X_n= E(X_{n+1}|\mathcal{F}_n)\). Apply \(\varphi\) to both sides such that \[\begin{eqnarray*} \varphi(X_n)&=& \varphi(E(X_{n+1}|\mathcal{F}_n))\\ &\le& E(\varphi(X_{n+1}|\mathcal{F}_n)).\mbox{ }\mbox{ }\mbox{ }(\mbox{Jensen's Inequality, }\varphi \mbox{ is convex}) \end{eqnarray*}\]

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